There is a renewed interest in the derivation of statistical mechanics from the dynamics of closed quantum systems. A central part of this program is to understand how closed quantum systems, i.e., in the absence of a thermal bath, initialized far-from-equilibrium can share a dynamics that is typical to the relaxation towards thermal equilibrium. Equilibration dynamics has been traditionally studied with a focus on the so-called quenches of large-scale many-body systems. We consider here the equilibration of a two-dimensional molecular model system describing the double proton transfer reaction in porphine. Using numerical simulations, we show that equilibration indeed takes place very rapidly (∼200 fs) for initial states induced by pump-dump laser pulse control with energies well above the synchronous barrier. The resulting equilibration state is characterized by a strong delocalization of the probability density of the protons that can be explained, mechanistically, as the result of (i) an initial state consisting of a large superposition of vibrational states, and (ii) the presence of a very effective dephasing mechanism.
There is a renewed interest in the derivation of statistical mechanics from the dynamics of closed quantum systems. A central part of this program is to understand how closed quantum systems, i.e., in the absence of a thermal bath, initialized far-from-equilibrium can share a dynamics that is typical to the relaxation towards thermal equilibrium. Equilibration dynamics has been traditionally studied with a focus on the so-called quenches of large-scale many-body systems. We consider here the equilibration of a two-dimensional molecular model system describing the double proton transfer reaction in porphine. Using numerical simulations, we show that equilibration indeed takes place very rapidly (∼200 fs) for initial states induced by pump-dump laser pulse control with energies well above the synchronous barrier. The resulting equilibration state is characterized by a strong delocalization of the probability density of the protons that can be explained, mechanistically, as the result of (i) an initial state consisting of a large superposition of vibrational states, and (ii) the presence of a very effective dephasing mechanism.
There is a renewed interest in the derivation of statistical mechanics from the dynamics of closed quantum systems. A central part of this program is to understand how far-from-equilibrium closed quantum system can behave as if relaxing to a stable equilibrium. Equilibration dynamics has been traditionally studied with a focus on the so-called quenches of large-scale many-body systems. Alternatively, we consider here the equilibration of a molecular model system describing the double proton transfer reaction in porphine. Using numerical simulations, we show that equilibration in this context indeed takes place and does so very rapidly (∼200fs) for initial states induced by pump-dump laser pulse control with energies well above the synchronous tunneling barrier.
Closed quantum systems evolve unitarily and therefore cannot converge in a strong sense to an equilibrium state starting out from a generic pure state. Nevertheless for large system size one observes temporal typicality. Namely, for the overwhelming majority of the time instants, the statistics of observables is practically indistinguishable from an effective equilibrium one. In this paper we consider the Loschmidt echo (LE) to study this sort of unitary equilibration after a quench. We draw several conclusions on general grounds and on the basis of an exactly solvable example of a quasifree system. In particular we focus on the whole probability distribution of observing a given value of the LE after waiting a long time. Depending on the interplay between the initial state and the quench Hamiltonian, we find different regimes reflecting different equilibration dynamics. When the perturbation is small and the system is away from criticality the probability distribution is Gaussian. However close to criticality the distribution function approaches a double-peaked, universal form.
We numerically investigate the relaxation dynamics in an isolated quantum system of interacting bosons trapped in a double-well potential after an integrability breaking quench. Using the statistics of the spectrum, we identify the postquench Hamiltonian as nonchaotic and close to integrability over a wide range of interaction parameters. We demonstrate that the system exhibits thermalization in the context of the eigenstate thermalization hypothesis (ETH). We also explore the possibility of an initial state to delocalize with respect to the eigenstates of the postquench Hamiltonian even for energies away from the middle of the spectrum. We observe distinct regimes of equilibration process depending on the initial energy. For low energies, the system rapidly relaxes in a single step to a thermal state. As the energy increases towards the middle of the spectrum, the relaxation dynamics exhibits prethermalization and the lifetime of the metastable states grows. Time evolution of the occupation numbers and the von Neumann entropy in the mode-partitioned system underpins the analyses of the relaxation dynamics.
The lack of knowledge about the detailed many-particle motion on the microscopic scale is a key issue in any theoretical description of a macroscopic experiment. For systems at or close to thermal equilibrium, statistical mechanics provides a very successful general framework to cope with this problem. However, far from equilibrium, only very few quantitative and comparably universal results are known. Here a quantum mechanical prediction of this type is derived and verified against various experimental and numerical data from the literature. It quantitatively describes the entire temporal relaxation towards thermal equilibrium for a large class (in a mathematically precisely defined sense) of closed many-body systems, whose initial state may be arbitrarily far from equilibrium.
Despite early developments on the foundations of quantum mechanics concern the wave function, quantum statistics has been developed with the density matrix formalism, leading to very important results in explaining molecular observations. Only recently, several authors argued a new interpretation by focusing on the wave function representing the quantum state of an isolated system, showing how a single wave function can exhibit statistical properties and generate the same results expected in the standard quantum statistical framework. Starting from these results, investigation on the foundations of quantum statistical mechanics has gained recently a renewed interest. As a matter of fact, the possibility of studying single molecule properties as well as the need of a better understanding of the effect of quantum dynamics, in order to develop new nanoscaled materials suitable to quantum computing tasks, have opened new intriguing questions leading to quantum statistical approaches far from being well understood and accepted. In this framework the behaviour of a single realization of quantum systems has gained a central role in the description of molecular systems. Furthermore, in recent years, an increasing number of studies has been presented on quantum dynamics through the numerical solution of the Schrödinger equation for systems of interacting components. These studies demonstrates that Quantum Dynamics Simulations could be a practicable route. In order to study phenomena such as dissipation, relaxation and thermalization, the focus has to be moved from isolated molecules to modular systems made of mutual interacting components, with model Hamiltonians possessing a sufficiently low dimensional representation. An important issue concerns the rules to be employed for the choice of the initial quantum state for the simulation of isolated systems. As long as one considers molecular degrees of freedom interacting with a (model) environment, there are no reasons to select a particular quantum state for the overall system and, therefore, a random choice has to be performed amongst a well defined statistical ensemble of pure states. Furthermore, one would like to operate a choice assuring that the simulation of the system is in a well defined thermal state with given temperature. This necessarily calls for a statistical description like for classical systems. A useful parametrization of the wave function will be presented in order to highlight the most appropriate variable for the statistical analysis. In particular some of them, called phases, retain all the dynamical information whereas the others, called populations, are the constants of motion. The latter, in particular are very important because they describe the equilibrium properties strictly related to the thermodynamic description. Since the dynamics of wave function does not supply any information about populations, the definition of a probability distribution on these variables is required. Different probability distributions on populations have been proposed, only on the bases of reasonable assumptions and they validation has been performed only with a posteriori considerations. In particular Fresch and Moro have demonstrated how the agreement with thermodynamics can be employed to discriminate different probability distributions on pure states. This had led the uniform ensembles to be, up to now, the most self-consistent models for quantum pure states. However, in this thesis I will highlight a drawback of the uniform distribution ensemble that can be described as follow: if we bring into contact two systems, even through a perturbative interaction, we are not able to describe the equilibrium properties after the interaction within the uniform distribution statistics, since the uniform character is lost. It represents a severe shortcoming of the statistical ensemble from a methodological point of view, since closed systems can be always considered as the result of interaction among previously isolated systems. On the other hand this drawback introduces a further requirement of a different nature that can be used for the definition of a new statistical ensemble. In this work I intend to find and characterize a statistical ensemble for populations that overcomes the drawbacks of the uniform distribution of pure states. The invariance of the thermal state in the coupling of identical systems will be used as a guideline in the definition af a new probability distribution on populations. Such an ensemble for pure states, called Thermalization Resilient Ensemble, provides a convenient framework for treating the interactions between quantum systems, as long as the structure of the statistical distribution is preserved and the identification of thermodynamic properties is assured. In perspective it should be the privileged statistical ensemble to implement Quantum Dynamics Simulations. Once the average properties of the Thermalization Resilient Ensemble have been introduced, I will obtain a probability distribution on pure states with the use of a geometrical analysis on the Hilbert space. The surface elements of an ellipsoidal manifold will be related to the probability density on populations. As a matter of fact the explicit form of the probability distribution is a prerequisite in order to perform Quantum Dynamical Simulations. However the results obtained through the geometrical analysis cannot be easily extended to systems with unbounded energy spectrum and an alternative strategy has been developed. A scaling algorithm on the basis of the uniform statistical ensemble will be described and this allows a well defined sampling of a probability distribution with desired averages. In this framework I demonstrate the emergence of thermodynamic behavior in the limit of macroscopic systems. In the last part of the thesis I consider the dynamical features of the thermalization experiment. Two identical systems, initially at different temperature, will be brought in interaction and the analysis of the final equilibrium state will be performed for two different generic forms of the interaction Hamiltonian, highlighting how the statistical approach can be very useful in the definition of the equilibrium in complex quantum systems.
Chemical reactions are commonly described by the reactive flux transferring the population from reactants to products across a double-well free energy barrier. Dynamics often involves barrier recrossing and quantum effects like tunneling, zero-point energy motion, and interference, which traditional rate theories, such as transition-state theory, do not consider. In this study, we investigate the feasibility of simulating reaction dynamics using a parametrically driven bosonic superconducting Kerr-cat device. This approach provides control over parameters defining the double-well free energy profile, as well as external factors like temperature and the coupling strength between the reaction coordinate and the thermal bath of nonreactive degrees of freedom. We demonstrate the effectiveness of this protocol by showing that the dynamics of proton-transfer reactions in prototypical benchmark model systems, such as hydrogen-bonded dimers of malonaldehyde and DNA base pairs, could be accurately simulated on the currently accessible Kerr-cat devices.